Sunday, January 8, 2012

Covered Call Are Naked Puts - The Proof

In my previous post I made the assertion that a covered call IS a naked put, synthetically. In this post I'll prove that with an example.

Firstly, it is important to know that every call option has a relationship with it's corresponding (same strike and expiry) put option as proven by and equation called the put/call parity equation. For your information, this equation goes like this: call price - corresponding put price = stock price - strike price + interest - dividend (if any).

It is not merely theoretical and we know this exists in the real traded market, because any anomalies in this relationship would be closed by arbitrageurs quicker than you can say 'Jack Robinson'. If for example call prices were too expensive compared to the corresponding put, arbitrageurs would sell the call, buy the put and buy the stock in equal proportions in a position called a conversion. This would guarantee a risk free profit. In reality, this is not achievable by a retail trader, so we can trust that option prices are in line with each other.

But I digress...

Let's suppose on Dec 15 2001 Tom the Trader wanted to collect some premium and had the view that GS would stay roughly where it is at around $92.50 or trickle up a bit until mid January 2012, so he sells 3 $90 Jan put contracts at $4.10, a face value of $27,000 (the amount Tom would have to cough up for the shares if assigned) and collecting $1230 in premium... a 'dangerous' naked put position.

If GS is above $90 as the January expiry date, Tom keeps his $1230 and pockets a tidy > 4% on the face value of the contracts (or > 12% on initial margin if you prefer looking at things that way {I don't}).

But what of the risk? What if Goldman has finally been caught out with one of its dirty deeds and the price collapses down to $70.00 by expiry? Well Tom could always cut his losses and exit before expiry, just like any trader can, but let's suppose Tom is a deer in the headlights and freezes?

Well Tom will most certainly be assigned the stock and will have paid face value for the stock, less the premium collected; a total of $25,770 for stock now worth $21,000. A loss of $4,770.

Ouch! Risky!

On the same day in Dec Tony the Trader thought it would be a great idea to do a covered call on GS, so buys 300 stock at $92.48 and sells 3 calls at $6.65. He pays $27,774 for the stock and collects $1,995 in call premium, bringing the cost base down to 25,779. Notice anything about similar about the put scenario yet?

If the stock closes anywhere at $90 or above at expiry, Tony will most certainly be assigned and will be paid $90 for the shares, collecting $27,000 ( a loss of $774) but will keep the $1995 call premium. This is an overall profit of $1,221.

But once again, what are the risks? Using our $70 at expiry scenario as we did with Tom's naked puts, Tony would be left holding stock now worth $21,000. He's lost $6,774 on the shares, but keeps the $1,995 call premium; an overall loss of  $4,779

Ouch! Risky also!

In each scenario the profit or loss of both traders is only $9 different to each other and this is accounted for by the interest accounted for in the option pricing model.

Pick any number at expiry and the profit or loss of these two scenarios will always be only $9 apart. Once interest for holding stock is accounted for, the result is the same. Because they are the same.

A covered call IS a (synthetic) naked put.

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